It implies that you are being cautious about not offending anyone, as if you are trying to avoid upsetting a bird by not disturbing its feathers.
Here are some examples:
* "I tried to be careful not to ruffle any feathers when I gave my presentation."
* "We'll need to be delicate in our approach so as not to ruffle any feathers."
* "The boss is very sensitive, so be careful not to ruffle any feathers."
It's important to note that the idiom is often used in a negative context, suggesting that someone is being too careful or overly concerned about upsetting others.